Mean, Median and Mode Year 8 Interactive Maths - Second Edition Mean, Median and Mode We use statistics such as the mean , median and mode to obtain information about a population from our sample set of observed values. Mean The mean (or average) of a set of data values is the sum of all of the data values divided by the number of data values. That is: Example 1 The marks of seven students in a mathematics test with a maximum possible mark of 20 are given below: 15 13 18 16 14 17 12 Find the mean of this set of data values. Solution: So, the mean mark is 15. Symbolically, we can set out the solution as follows: So, the mean mark is 15. Median The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value.

Example 2 The marks of nine students in a geography test that had a maximum possible mark of 50 are given below: 47 35 37 32 38 39 36 34 35 Find the median of this set of data values. Solution: Arrange the data values in order from the lowest value to the highest value: 32 34 35 35 36 37 38 39 47 The fifth data value, 36, is the middle value in this arrangement. Note: In general: If the number of values in the data set is even, then the median is the average of the two middle values. Example 3 Find the median of the following data set: 12 18 16 21 10 13 17 19 Solution: Arrange the data values in order from the lowest value to the highest value: 10 12 13 16 17 18 19 21 The number of values in the data set is 8, which is even.

So, the median is the average of the two middle values. Alternative way: There are 8 values in the data set. The fourth and fifth scores, 16 and 17, are in the middle. That is, there is no one middle value. Note: Half of the values in the data set lie below the median and half lie above the median. The median is the most commonly quoted figure used to measure property prices. The use of the median avoids the problem of the mean property price which is affected by a few expensive properties that are not representative of the general property market. Mode The mode of a set of data values is the value(s) that occurs most often. The mode has applications in printing. For example, it is important to print more of the most popular books; because printing different books in equal numbers would cause a shortage of some books and an oversupply of others. Likewise, the mode has applications in manufacturing. For example, it is important to manufacture more of the most popular shoes; because manufacturing different shoes in equal numbers would cause a shortage of some shoes and an oversupply of others.

Example 4 Find the mode of the following data set: 48 44 48 45 42 49 48 Solution: The mode is 48 since it occurs most often. Note: It is possible for a set of data values to have more than one mode. If there are two data values that occur most frequently, we say that the set of data values is bimodal. If there is no data value or data values that occur most frequently, we say that the set of data values has no mode. Analysing Data The mean , median and mode of a data set are collectively known as measures of central tendency as these three measures focus on where the data is centred or clustered. To analyse data using the mean, median and mode, we need to use the most appropriate measure of central tendency. The following points should be remembered: The mean is useful for predicting future results when there are no extreme values in the data set. However, the impact of extreme values on the mean may be important and should be considered. E.g. The impact of a stock market crash on average investment returns.

The median may be more useful than the mean when there are extreme values in the data set as it is not affected by the extreme values. The mode is useful when the most common item, characteristic or value of a data set is required. Key Terms Grade 5: Data and Graphs: Overview 8 1 Notice that in the stem-and-leaf plot both the tens and ones digits are in order from least to greatest. Where parts of the display have no data—no one was in their 20s or 60s for example—there is a gap. Where many data are close together—ages under ten and ages in the 30s—there are clusters. Stem-and-leaf plots help us order data and show a great deal of data in a small space. Two sets of data may be shown on a double bar graph. For example, at the family reunion, it was found that four surnames (Brown, Kildare, Roberts, and O'Hara) were represented. Males and females of each family name could be shown in a double bar graph such as this one.

More data about people at the family reunion could be shown in a frequency table and histogram, both of which use 10-year intervals for ages. In the frequency table, tallies show the numbers of people in various age groups. Frequency Table 80 – 89 1 The same data is shown on a histogram, a display that is like a bar graph except that it shows interval data. With numerical data, the terms mean, median, and mode tell us about a typical value. The range tells us how the data is “spread out,” or the difference between the greatest and the least numbers in a set of data. Suppose that the following numbers show the number of miles driven by five families who came to the family reunion: 5, 120, 36, 36, 97, 509, 247 Put the numbers in order. It's easy to see that the numbers of miles vary from 5 to 509. The range is the difference in the least and greatest number: 509 − 5 = 504. The mode is the number that occurs most often.

Sets of data may have more than one mode, or they may have no mode if all the numbers are different. The mode of this data set is 36. The median is the middle value in a set of data in which the data are arranged in order. If there is an even number of data, the median is the mean, or average, of the middle two numbers. Following are the numbers in order: 5, 36, 36, 97, 120, 247, 509 Since there are 7 numbers, the fourth (97) is the middle number, or the median. The mean is the average of the numbers. To find the mean, add all the numbers and divide by the number of addends. The mean of the numbers is 1,050 ÷ 7 = 150. Another form of graph, the line plot, makes it easy to see the mean, median, mode, and range of a set of data. The line plot uses a number line and Xs to represent each number.

This line plot shows these scores from the game of horseshoes at the family reunion: 18, 14, 22, 33, 7, 7, 30, 34, 30, 10, 15. The data has two modes—7 and 30. The line plot lets you see these easily, because the numbers are represented by Xs. The range is34 − 7 or 27. The median, the middle value is the sixth X, or 18. To find the mean, add the scores and divide by the number of scores (220 ÷ 11 = 20). Line graphs show change over time. Broken lines show increases or decreases in data, and the slope of the lines tells whether the change is gradual or rapid. Horizontal line segments show periods of no change. This graph shows temperatures over an 8-hour period at the family reunion. How can the graph be interpreted? The temperature rose slowly from 8:00 to 9:00 a.m., and then rose more rapidly from 9:00 until 10:00 a.m.

and from 10:00 until 11:00 a.m. The change from 11:00 a.m. until noon was a more gradual increase. The temperature remained constant from noon until 1:00 p.m.. and also from 3:00 to 4:00 p.m. It gradually fell from 1:00 p.m.. until 3:00 p.m. Double line graphs let people compare two sets of data over time. Often they use two colors and a code to show the sets of data. Here is a double line graph showing temperatures in indoor and outdoor areas at the family reunion. Even a quick look at the graph shows more variation or "ups and downs" on outdoor temperatures than indoor temperatures. You can see that the indoor temperature, about 72°F, was higher than the outdoor temperature at 8:00 a.m. As the day progressed, the indoor temperature rose to 75°F, then stayed constant from noon until 3:00 p.m., when it fell 2 degrees, then rose again to 74°F at 4:00 p.m.

You can visually find the greatest difference in temperatures at noon—a difference of about 12 degrees. With so many types of graphs available, students must learn how to choose appropriate types. Bar graphs are often a good choice to show comparisons among data. Histograms show data that is organized in equal intervals. Line graphs and double line graphs are well suited to showing change over time. Pictographs are appropriate when the data are multiples of a number—and where pictures help to convey information. Circle graphs are well suited for showing parts of a whole. Definition and examples Measures of Central Tendency | define Measures of Central Tendency - Statistics -Free Math Dictionary Online Measures of Central Tendency Definition of Measures of Central Tendency A measure of central tendency is a measure that tells us where the middle of a bunch of data lies. The three most common measures of central tendency are the mean, the median, and the mode. More About Measures of Central Tendency Mean: Mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average. Median: Median is the number present in the middle when the numbers in a set of data are arranged in ascending or descending order.

If the number of numbers in a data set is even, then the median is the mean of the two middle numbers. Mode: Mode is the value that occurs most frequently in a set of data. Examples of Measures of Central Tendency For the data 1, 2, 3, 4, 5, 5, 6, 7, 8 the measures of central tendency are Mean = Median, Quartiles, Percentiles (examples, solutions, videos) Find the median, lower quartile and upper quartile of the following numbers. 12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25 Solution: First, arrange the data in ascending order: Median (middle value) = 22 Lower quartile (middle value of the lower half) = 12 Upper quartile (middle value of the upper half) = 36 If there is an even number of data items, then we need to get the average of the middle numbers. Example: Find the median, lower quartile, upper quartile, interquartile range and range of the following numbers.

12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25, 65 Solution: First, arrange the data in ascending order: Lower quartile or first quartile = Median or second quartile = Upper quartile or third quartile = Interquartile range = Upper quartile – lower quartile = 39 – 13 = 26 Range = largest value – smallest value = 65 – 5 = 60 When evaluating the quartiles, always remember to first arrange the data in increasing order. Videos Quartiles Year 9 Interactive Maths - Second Edition Quartiles If a data set of values is arranged in ascending order of magnitude, then: The median is the middle value of the data set. The lower quartile (Q1) is the median of the lower half of the data set. The upper quartile (Q3) is the median of the upper half of the data set. The interquartile range (IQR) is the spread of the middle 50% of the data values.

So: The interquartile range is a more useful measure of spread than the range as it describes the middle 50% of the data values. Example 8 Find the median, lower quartile, upper quartile and interquartile range of the following data set of scores: 18 20 23 20 23 27 24 23 29 Solution: Arrange the values in ascending order of magnitude: 18 20 20 23 23 23 24 27 29 There are 9 values in the data set. This means the middle 50% of the data values range from 20 to 25.5. Note: The quartiles divide the set of measurements into four equal parts. Twenty-five per cent of the measurements are less than the lower quartile , fifty per cent of the measurements are less than the median and seventy-five per cent of the measurements are less than the upper quartile . So, fifty per cent of the measurements are between the lower quartile and the upper quartile.

The lower quartile, median and upper quartile are often denoted by Q1, Q2 and Q3 respectively. The median is also denoted by m. Key Terms Median - Definition, Formula & Examples | MathCaptain.com Median Median Median is a measure which is easy to understand and easy to determine. It is the halfway point in a data set and hence known as the positional average. Since it is not dependent on the actual values in the data set, it is less affected by extreme values. Against these merits, median is not based on all values and is not capable of algebraic treatments. It is also affected by sampling fluctuations. Median is also a measure of position along with quartiles and Deciles. Median can also be understood as the second quartile or 50th percentile. Definition If the data values of a numerical data set is arranged in ascending or descending order of magnitude, the middle most item is called the median of the data set. In numerical data distributions one half of the items are less than the median value and the other half is greater than the median value. The symbol used for median is MD. In the data set displayed above, the median is 7.

It can be seen on either side of mean there are 6 values. Median Statistics Formula The following two observations lead to the formula for median: 1) When the number of values in the data set is odd, the median is the middle value when the data is ordered. Thus the median is the $(\frac{n+1}{2})^{th}$ observation. 2) When the number values in the data set is even, there are two middle values in the ordered array and median is the average of these two middle values. Thus the median is the average of the $(\frac{n}{2})^{th}$ and $(\frac{n}{2}+1)^{th}$ observations. When the data is given in frequency distribution the median class is found by using the above formula. The actual value of the median is then determined proportionately. Examples Solved Examples Question 1: The ages of 15 old people living in an old age Home are 65, 62, 78, 82, 89, 90, 73, 69, 70, 70, 71, 72, 78, 68 and 72.

Find the median of the data set. Solution: Statistics: Power from Data! Calculating the median Comparing the mean and median If observations of a variable are ordered by value, the median value corresponds to the middle observation in that ordered list. The median value corresponds to a cumulative percentage of 50% ( i.e. , 50% of the values are below the median and 50% of the values are above the median). The position of the median is {(n + 1) ÷ 2}th value, where n is the number of values in a set of data. In order to calculate the median, the data must first be ranked (sorted in ascending order). The median is the number in the middle. Median = the middle value of a set of ordered data. The median is usually calculated for numeric variables , but may also be calculated for categorical variables that are sequenced, such as the categories in a satisfaction survey: excellent, good, satisfactory and poor. These qualitative categories can be ranked in order, and thus, are considered ordinal . Raw data In raw data , the median is the point at which exactly half of the data are above and half below. These halves meet at the median position.

If the number of observations is odd, the median fits perfectly and the depth of the median position will be a whole number. If the number of observations is even, the depth of the median position will include a decimal. You need to find the midpoint between the numbers on either side of the median position. Example 1 – Raw data (discrete variables) Imagine that a top running athlete in a typical 200-metre training session runs in the following times: 26.1, 25.6, 25.7, 25.2 et 25.0 seconds. How would you calculate his median time? First, the values are put in ascending order: 25.0, 25.2, 25.6, 25.7, 26.1. Then, using the following formula, figure out which value is the middle value. Remember that n represents the number of values in the data set. Median = {(n + 1) ÷ 2}th value = (5 + 1) ÷ 2 = 3 The third value in the data set will be the median.

Since 25.6 is the third value, 25.6 seconds would be the median time. = 25.6 secondes Example 2 – Raw data (discrete variables) Now, if the runner sprints the sixth 200-metre race in 24.7 seconds, what is the median value now? Again, you first put the data in ascending order: 24.7, 25.0, 25.2, 25.6, 25.7, 26.1. Then, you use the same formula to calculate the median time. Median = {(n + 1) ÷ 2}th value = (6 + 1) ÷ 2 = 7 ÷ 2 = 3,5 Since there is an even number of observations in this data set, there is no longer a distinct middle value. The median is the 3.5th value in the data set meaning that it lies between the third and fourth values. Thus, the median is calculated by averaging the two middle values of 25.2 and 25.6. Use the formula below to get the average value.

Average = (value below median + value above median) ÷ 2 = (third value + fourth value) ÷ 2 = (25.2 + 25.6) ÷ 2 = 50.8 ÷ 2 = 25.4 The value 25.4 falls directly between the third and fourth values in this data set, so 25.4 seconds would be the median time. Ungrouped frequency distribution In order to find the median using cumulative frequencies (or the number of observations that lie above or below a particular value in a data set), you must calculate the first value with a cumulative frequency greater than or equal to the median. If the median's value is exactly 0.5 more than the cumulative frequency of the previous value, then the median is the midpoint between the two values.

Example 3 – Ungrouped frequency table (discrete variables) Imagine that your school baseball team scores the following number of home runs in 10 games: 4, 5, 8, 5, 7, 8, 9, 8, 8, 7 If you were to place the total home runs in a frequency table, what would the median be? First, put the scores in ascending order: 4, 5, 5, 7, 7, 8, 8, 8, 8, 9 Then, make a table with two columns. Label the first column "Number of home runs" and then list the possible number of home runs the team could get. You can start from 0 and list up until the number 10, but since the team never scored less than 4 home runs, you may wish to start listing at the number 4. Label the second column "Frequency." In this column, record the number of times 4 home runs were scored, 5 home runs were scored and so on. In this case, there was only one time that 4 home runs were scored, but two times that 5 home runs were scored.

If you add all of the numbers in the Frequency column, they should equal 10 (for the 10 games played). Table 1. Number of home runs in 10 baseball games Number of home runs (x) Frequency (f) To find the median, again use the same formula: Median = {(n + 1) ÷ 2}th value = (10 + 1) ÷ 2 = 5.5 = the median is the 5.5th value in the data set To get the median, add up the numbers in the Frequency column until you get to 5 (and since the total number of games is 10, the remaining numbers in that column should also equal 5). You will reach 5 after adding all of the frequencies up to and including those for the 7 home runs. The next set of five will begin with the frequencies for 8 home runs. The median (the 5.5th value) lies between the fifth value and the sixth value. Thus, the median lies between 7 home runs and 8 home runs.

If you calculate the average of these values (using the same formula used in Example 2), the result is 7.5. Average = (middle value before + middle value after) ÷ 2 = (fifth value + sixth value) ÷ 2 = (7 + 8) ÷ 2 = 15 ÷ 2 = 7.5 Technically, the median should be a possible variable. In the above example, the variables are discrete and always whole numbers. Therefore, 7.5 is not a possible variable—no one can hit 7 and a half home runs. Thus, this number only makes sense statistically. Some mathematicians may argue that 8 is a more appropriate median. Grouped frequency distribution Sometimes it does not make sense to list each individual variable when a frequency distribution table would be long and cumbersome to work with. In order to simplify this, divide the range of data into intervals and then list the intervals in a frequency distribution table, including a column for the cumulative percentage. (For more information, refer to the Cumulative frequency section.) The calculation to find the median is a little longer because the data have been grouped into intervals and, therefore, all of the original information has been lost.

Some textbooks simply take the midpoint of the interval as the median. However, that method is an over-simplification of the true value. Use the following calculations to find the median for a grouped frequency distribution. Figure out which interval contains the median by using the (n + 1) ÷ 2 formula. Take whatever value the calculation gives you and then add up the numbers in the frequency column until you come to that value (just like Example 3). For example, if your median is the 13.5th value, add up the frequencies until you come to the 13th and 14th values. Whichever interval contains these values is called the median group. Find the cumulative percentage of the interval preceding the median group. Label this value A. Using this cumulative percentage, calculate how many numbers are needed in order to add up to 50% of the total cumulative percentage. This value will be labeled B. Use the following formula to calculate B: B = 50 - A Figure out the range (how many numbers the interval covers). Call this value C. Then, find the percentage for the median interval. Call this value D.

Calculate how many data values you have to count in the median group to get 50% of the total data set by using the following formula. Call this value E. E = (B ÷ D) x C Find out what the median value is by adding the value for E to the lower value of the median interval: Median = lower value + E Since E = (B ÷ D) x C, this formula can also be written as: Median = lower value + (B ÷ D) x C If the cumulative frequency for an interval is exactly 50%, then the median value would be the endpoint of this interval. Let's make this clear with an example! Example 4 – Grouped variables - frequency distribution (continuous or discrete) Using the same information from Example 4 in the Mean section, imagine that you surveyed 50 Grade 10 girls to find out how tall each one is in centimetres. After gathering all of your data, you created a frequency distribution table that looked like this: Table 2. Height of Grade 10 girls Height (cm) 8 100 Using the grouped data, you created a cumulative frequency graph to accompany your table.

The endpoints of the height intervals, the numbers for cumulative frequency and the numbers for cumulative percentage have been plotted on the graph. Top of Page By just looking at the graph, you can try to find the median value. The median is the point where the x-axis (Height) intersects with the midpoint (25) of the y-axis (Cumulative frequency). You will see that the median value is approximately 164 cm. Using mathematical calculations, you can find out that the value is actually 163.9 cm. Here's how: According to the information provided in Table 2: Median = {(n + 1) ÷ 2}th value = (50 + 1) ÷ 2 6 6 7 8 50 *Note: The stems have been split into smaller intervals. Stem 15(0) means that all the data fall within the interval 150 to 154. Stem 15(5) means that the data are in the interval of 155 to 159. There are 50 pieces of data, so the median is the value of the 25.5th observation.

Median = {(n + 1) ÷ 2}th value = (50 + 1) ÷ 2 = 51 ÷ 2 = 25.5 Therefore, the median lies between the 25th and 26th values. To find out what these values are, count each value in the Leaf column until you have reached the 25th and 26th values. These values lie in the 16(0) interval, meaning the 160 to 164 interval. The numbers in the leaf column represent the numbers in the interval ( e.g. , 3 represents 163). Thus, the median lies between 163 cm (25th value) and 164 cm (26th value). The median is found by averaging these two values. Average = (value before median + value after median) ÷ 2 = (25th value + 26th value) ÷ 2 = (163 + 164) ÷ 2 = 163.5 Since height is a continuous variable, 163.5 cm is an acceptable median value.

The median obtained from the cumulative frequency graph (164 cm) is not the same value as the median obtained from the calculation in Example 4 (163.9 cm) or from the stem and leaf plot (163.5 cm). This is because you can only find an approximation for the median, unless the graph is drawn precisely with all the information used. The calculations in Example 4 are only approximations, since grouped data do not tell you how the 36% of the 50 girls found in the median interval are distributed within the interval. As a result, we make the assumption that they are uniformly distributed, and this may lead to a slightly different median. However, the stem and leaf plot is the most accurate means of obtaining a median because it uses all of the actual values. Comparing the mean and median It is possible for the mean and median of a distribution to have the same value. This is always the case if distribution is symmetrical as in a normal distribution . If the distribution is roughly symmetrical, then the two values will be close together.

In the example of the heights of the 50 Grade 10 girls, the mean (164.5 cm) is very close to the value of the median (163.5 cm). This is because the distribution is roughly symmetrical (see the stem and leaf plot in the above example ). However, one number can alter the mean without affecting the median. Example 6 – Comparing the mean and median Consider the following sets of data that represent the number of points scored by 3 players in 11 lacrosse games.

Eileen: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3 Mean = 22 ÷ 11 = 2 Jeremy: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4 Mean = 23 ÷ 11 = 2.1 Randy: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 14 Mean = 33 ÷ 11 = 3 Median = 2 The three sets of data above are identical except for the last observation values (3, 4 and 14). The median does not alter because it is only dependent on the middle observation's value. The mean does change, however, because it is dependent on the average value of all observations. So, in the above example, as the last value of the last observation increases, so too does the mean. In the third data set, the value of 14 is very different from any other values.

When an observation is very different from all other observations in a data set, it is called an outlier . (For more information on outliers, see the Stem and leaf plots section.) The mean is the measure of central tendency most affected by outliers. Outliers can sometimes occur as a result of error or deliberate misinformation. In these cases, the outliers should be excluded from the measure of central tendency. Other times, outliers just show how different one value is, and this can be a very useful piece of data. Example 7 – Comparing the mean and median When house prices are referred to in newspapers, generally the median price is quoted. Why is this measure used instead of the mean? There are many moderately priced houses, but there are also some expensive ones and a few very expensive ones. The mean figure could be quite high as it includes the prices of the more expensive houses. But the median gives a more accurate and realistic value of the prices faced by most people. In summary, the median is the central number and is good to use in skewed (or unbalanced) distributions because it is not affected by outliers. Example 8 – Comparing the mean and median Suppose you want to know how much money a family could afford to spend on housing.

This would depend on the total family income. For a family of five (two parents who work and three children with no income) the mean income of each family member is the total income divided by five ( e.g. , 60,000 ÷ 5 = 12,000). However, the median income would be zero, because more than half of the members of the family make nothing. In some situations, the mean can be much more informative than the median. Example 9 – Comparing the mean and median If you want to find out whether a country is wealthy or not, you might consider using the median as your measure of central tendency instead of the mean. The mean family income could be quite high if income is highly concentrated in a few very wealthy families (despite the fact that most families might earn essentially nothing). Thus, the median family income would be a more meaningful measure—at least half the families would earn the median income or less, and at least half would earn at least as much as the median income or more.

Mean, Mode and Median - Measures of Central Tendency - When to use with Different Types of Variable and Skewed Distributions | Laerd Statistics Measures of Central Tendency Introduction A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode. The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used. Mean (Arithmetic) The mean (or average) is the most popular and well known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous data (see our Types of Variable guide for data types).

The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if we have n values in a data set and they have values x1, x2, ..., xn, the sample mean, usually denoted by (pronounced x bar), is: This formula is usually written in a slightly different manner using the Greek capitol letter, , pronounced "sigma", which means "sum of...": You may have noticed that the above formula refers to the sample mean. So, why have we called it a sample mean? This is because, in statistics, samples and populations have very different meanings and these differences are very important, even if, in the case of the mean, they are calculated in the same way. To acknowledge that we are calculating the population mean and not the sample mean, we use the Greek lower case letter "mu", denoted as µ: The mean is essentially a model of your data set. It is the value that is most common. You will notice, however, that the mean is not often one of the actual values that you have observed in your data set.

However, one of its important properties is that it minimises error in the prediction of any one value in your data set. That is, it is the value that produces the lowest amount of error from all other values in the data set. An important property of the mean is that it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero. When not to use the mean The mean has one main disadvantage: it is particularly susceptible to the influence of outliers. These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. For example, consider the wages of staff at a factory below: Staff 90k 95k The mean salary for these ten staff is $30.7k. However, inspecting the raw data suggests that this mean value might not be the best way to accurately reflect the typical salary of a worker, as most workers have salaries in the $12k to 18k range. The mean is being skewed by the two large salaries.

Therefore, in this situation, we would like to have a better measure of central tendency. As we will find out later, taking the median would be a better measure of central tendency in this situation. Another time when we usually prefer the median over the mean (or mode) is when our data is skewed (i.e., the frequency distribution for our data is skewed). If we consider the normal distribution - as this is the most frequently assessed in statistics - when the data is perfectly normal, the mean, median and mode are identical. Moreover, they all represent the most typical value in the data set. However, as the data becomes skewed the mean loses its ability to provide the best central location for the data because the skewed data is dragging it away from the typical value. However, the median best retains this position and is not as strongly influenced by the skewed values. This is explained in more detail in the skewed distribution section later in this guide. Join the 10,000s of students, academics and professionals who rely on Laerd Statistics. TAKE THE TOUR PLANS & PRICING Median The median is the middle score for a set of data that has been arranged in order of magnitude. The median is less affected by outliers and skewed data.

In order to calculate the median, suppose we have the data below: 65 87 89 Only now we have to take the 5th and 6th score in our data set and average them to get a median of 55.5. Mode The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram. You can, therefore, sometimes consider the mode as being the most popular option. An example of a mode is presented below: Normally, the mode is used for categorical data where we wish to know which is the most common category, as illustrated below: We can see above that the most common form of transport, in this particular data set, is the bus. However, one of the problems with the mode is that it is not unique, so it leaves us with problems when we have two or more values that share the highest frequency, such as below: We are now stuck as to which mode best describes the central tendency of the data. This is particularly problematic when we have continuous data because we are more likely not to have any one value that is more frequent than the other.

For example, consider measuring 30 peoples' weight (to the nearest 0.1 kg). How likely is it that we will find two or more people with exactly the same weight (e.g., 67.4 kg)? The answer, is probably very unlikely - many people might be close, but with such a small sample (30 people) and a large range of possible weights, you are unlikely to find two people with exactly the same weight; that is, to the nearest 0.1 kg. This is why the mode is very rarely used with continuous data. Another problem with the mode is that it will not provide us with a very good measure of central tendency when the most common mark is far away from the rest of the data in the data set, as depicted in the diagram below: In the above diagram the mode has a value of 2. We can clearly see, however, that the mode is not representative of the data, which is mostly concentrated around the 20 to 30 value range. To use the mode to describe the central tendency of this data set would be misleading.

PLANS &AMP; PRICING Skewed Distributions and the Mean and Median We often test whether our data is normally distributed because this is a common assumption underlying many statistical tests. An example of a normally distributed set of data is presented below: When you have a normally distributed sample you can legitimately use both the mean or the median as your measure of central tendency. In fact, in any symmetrical distribution the mean, median and mode are equal. However, in this situation, the mean is widely preferred as the best measure of central tendency because it is the measure that includes all the values in the data set for its calculation, and any change in any of the scores will affect the value of the mean. This is not the case with the median or mode. However, when our data is skewed, for example, as with the right-skewed data set below: we find that the mean is being dragged in the direct of the skew. In these situations, the median is generally considered to be the best representative of the central location of the data. The more skewed the distribution, the greater the difference between the median and mean, and the greater emphasis should be placed on using the median as opposed to the mean.

A classic example of the above right-skewed distribution is income (salary), where higher-earners provide a false representation of the typical income if expressed as a mean and not a median. If dealing with a normal distribution, and tests of normality show that the data is non-normal, it is customary to use the median instead of the mean. However, this is more a rule of thumb than a strict guideline. Sometimes, researchers wish to report the mean of a skewed distribution if the median and mean are not appreciably different (a subjective assessment), and if it allows easier comparisons to previous research to be made. Summary of when to use the mean, median and mode Please use the following summary table to know what the best measure of central tendency is with respect to the different types of variable . Type of Variable The Median of a Set of Data Unit 8 > Lesson 4 of 8 Problem: The Doran family has 5 children, aged 9, 12, 7, 16 and 13. What is the age of the middle child?

Solution: Ordering the childrens' ages from least to greatest, we get: 7, 9, 12, 13, 16 Answer: The age of the middle child is the middlemost number in the data set, which is 12. In the problem above, we found the median of a set of 5 numbers. Definition: The median of a set of data is the middlemost number in the set. The median is also the number that is halfway into the set. To find the median, the data should first be arranged in order from least to greatest. To remember the definition of a median, just think of the median of a road, which is the middlemost part of the road. In the problem above, 12 is the median: it is the number that is halfway into the set. There are two children older than 12 and two children younger than 12. Let's look at some more examples. Example 1: The Jameson family drove through 7 states on their summer vacation. Gasoline prices varied from state to state. What is the median gasoline price?

$1.79, $1.61, $1.96, $2.09, $1.84, $1,75, $2.11 Solution: Ordering the data from least to greatest, we get: $1.61, $1.75, $1.79, $1.84, $1.96, $2.09, $2.11 Answer: The median gasoline price is $1.84. (There were 3 states with higher gasoline prices and 3 with lower prices.) Example 2: During the first marking period, Nicole's math quiz scores were 90, 92, 93, 88, 95, 88, 97, 87, and 98. What was the median quiz score? Solution: Ordering the data from least to greatest, we get: 87, 88, 88, 90, 92, 93, 95, 96, 98 Answer: The median quiz score was 92.

(Four quiz scores were higher than 92 and four were lower.) In each of the examples above, there is an odd number of items in each data set. In Example 1, there are 7 numbers in the data set; in Example 2 there are 9 numbers. Let's look at some examples in which there is an even number of items in the data set. Example 3: A marathon race was completed by 4 participants. What was the median race time? 2.7 hr, 8.3 hr, 3.5 hr, 5.1 hr Solution: Ordering the data from least to greatest, we get: 2.7, 3.5, 5.1, 8.3 Since there is an even number of items in the data set, we compute the median by taking the mean of the two middlemost numbers. 3.5 + 5.1 = 8.6 The median race time was 4.3 hr. Example 4: The salaries of 8 employees who work for a small company are listed below. What is the median salary?

$40,000, $29,000, $35,500, $31,000, $43,000, $30,000, $27,000, $32,000 Solution: Ordering the data from least to greatest, we get: $27,000, $29,000, $30,000, $31,000, $32,000, $35,500, $40,000, $43,000 Since there is an even number of items in the data set, we compute the median by taking the mean of the two middlemost numbers.

Question: If a set of mathematical data is arranged in order, what name is given to the middle value?
Answer: Median